iFirst (2012) 1–13
Desalination and Water Treatment www.deswater.com 1944-3994/1944-3986 Ó 2012 Desalination Publications. All rights reserved doi: 10.1080/19443994.2012.704718
Parametric equations for the variables of a steady-state model of a multi-effect desalination plant Patricia Palenzuela*, Diego Alarco´n, Guillermo Zaragoza, Julia´n Blanco, Mercedes Ibarra
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CIEMAT-Plataforma Solar de Almerı´a, Ctra. de Sene´s s/n, 04200 Tabernas, Almerı´a, Spain Tel. +34 950387909; Fax: +34 950365015; email:
[email protected] Received 8 April 2012; Accepted 15 June 2012
ABSTRACT
In this work a steady-state model is developed of an MED plant. Its development and validation have been carried out by experimental data obtained from an MED pilot plant located at the Plataforma Solar de Almerı´a (PSA), in the southeast of Spain. It is a vertical arrangement forward-feed MED plant with preheaters, which uses hot water as the thermal energy source. In order to run the model, a series of parametric equations have been determined for the following variables: the overall heat transfer coefficient for the first effect (Uh), the overall heat transfer coefficient for the preheaters (Up(i)), the vapor temperature inside the first effect (Tv(1)), and the cooling seawater outlet temperature (Tcwout). They have been obtained from a three-level factorial experimental design (3k), performing a total of 81 experiments (34). The results obtained showed a good fit to the estimated models for the response variables. Keywords: Solar desalination; Multi-effect distillation; Modeling
1. Introduction Industrial desalination of seawater is one of the possible solutions to alleviate the worldwide scarcity of freshwater. The industrial desalination processes can be split into two main categories: (1) thermal processes that include multi-stage flash (MSF) and multi-effect distillation (MED) as the most commercially successful and (2) membrane processes including reverse osmosis (RO). The advantages of thermal desalination processes are their ability to be driven by low energy thermal sources, their reliability, easier operation and maintenance and high purity freshwater. Among the thermal desalination processes, MED has the highest thermal efficiency and the lowest power consumption [1]. *Corresponding author.
However, the disadvantage of thermal desalination processes is that the overall energy consumption is high, so the use of renewable energies is required to guarantee the sustainability of this technological option [2,3]. The usual coincidence in many locations of fresh water shortage, abundant seawater resources and high isolation levels makes thermal seawater desalination driven by solar energy as one of the most promising processes to obtain fresh water. This can be possible by the coupling of a conventional thermal distillation plant with a solar thermal system [4]. The improvement in the design of a MED process can be reached by the prediction of its performance over a wide range of operating conditions. Mathematical modeling and computer simulations can provide an insight into the workings of the system and help to
Presented at the International Conference on Desalination for the Environment, Clean Water and Energy, European Desalination Society, 23–26 April 2012, Barcelona, Spain
P. Palenzuela et al. / Desalination and Water Treatment
understand in detail the process elements in order to predict the behavior and the efficiency of the system. Many articles have been published on the topic of modeling and simulation of MED processes. El-Dessouky et al. have made several contributions in the mathematical modeling of MED plants [5,6]. El-Nashar et al. [7,8] presented a mathematical simulation of the steady-state operation of a solar MED unit, which was validated with experimental data from a pilot plant located at Abu Dhabi, UAE. A computer simulation model of a multi-effect thermal vapor compression (TVC-MED) was presented by Kamali [9] to predict the influence of all factors on heat transfer coefficients, temperature and pressure, total capacity and performance ratio of the system under design and operating conditions. Also, Trostmann [10] developed a model and carried out steady-state simulations of a MED-TVC, where heat transfer coefficients correlations for condensation and evaporation in the tubes were discussed and compared. Gautami et al. [11] developed mathematical models in order to select the optimal configuration of a MED process based on the product concentration and steam economy. Finally, the authors developed a model of an MED plant, based on the MED pilot plant located at the Plataforma Solar de Almerı´a (PSA) [12]. To solve this model, a parameterization of the overall heat transfer coefficients was carried out by performing a series of experiments at the MED-PSA plant. The correlations obtained were based on the characterization published by El-Nashar [8]. In this study, new correlations have been obtained by the authors by means of a parametric study. The parametric equations have been determined for the following variables: the overall heat transfer coefficient for the first effect (Uh), the overall heat transfer coefficient for the preheaters (Up(i)), the vapor temperature inside the first effect (Tv(1)), and the cooling seawater outlet temperature (Tcw,out). Such parametric equations have been obtained using a factorial design.
saturated steam (70˚C, 0.31 bars). The hot water for the first cell is provided either with a solar field composed of static compound parabolic concentrators (CPC) or with a double-effect absorption heat pump, DEAHP (LiBr-H2O), which was manufactured by ENTROPIE in 2005 in the framework of the AQUASOL project [16– 18]. In the modelling developed in this study, only the case in which the hot water comes from the solar filed has been taken into account. The flow sheet of the process is shown in Fig. 1. Firstly, the pumped feed seawater flows through the condenser and the preheaters of the plant in order to preheat it before reaching the first effect. Once in the first effect, the seawater is sprayed through a spraying
Mf, Tf Solar field
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2
Th (1), Mh
Pv (1)
Th (2) Pv (2)
Pv (4)
Pv (6)
Pv (8)
Pv (10)
Pv (12)
Feed seawater Mf, Tf
2. Description of the plant Pv (14)
2.1. Process design The desalination plant at the PSA, with 14 cells or effects, is a forward-feed MED unit manufactured and delivered by ENTROPIE in 1987 [13]. The cells are in a vertical arrangement at decreasing pressures from cell 1 to cell 14 [14]. The first cell, which was modified in 2005 within the framework of a project called AQUASOL, works with hot water as the heat transfer media [15]. Originally, the first cell worked with low-pressure
Pvc
Condenser
Brine reject Mb, Tb
Cooling Distillate water Feed and cooling seawater production seawater Mcw, Tcw(2) M d, T d (Mf +Mcw), Tcw(1)
Fig. 1. Schematic diagram of the multi-effect distillation plant at the Plataforma Solar de Almerı´a. Dashed lines show the steam flow; solid lines the distillate flow; Ushaped lines represent the brine flow from one effect to the next.
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tray falling over a horizontal-tube bundle. Then, it is built on thin falling film that coats the surface of the tubes entirely. On the other hand, the hot water flows inside the tubes and releases its sensible heat to the seawater, evaporating part of it. Vapor generated in the first effect flows to the preheater located next to it, through a wire mesh demister that removes the entrained brine droplets. Here, vapor transfers part of its latent heat to the seawater which is circulating inside the preheater bundle tube, increasing its temperature. As a consequence, a small amount of the vapor condenses, which is the first distillate of the process. The vapor that has not condensed flows through the inside of the second effect tube bundle, being the thermal energy source in the evaporation–condensation process in this effect. Then, the vapor condenses by transferring its latent heat to the more concentrated brine flowing from the previous effect. The same process is repeated in the rest of effects, being the vapor produced in the previous effect the thermal energy source of the effect. In each effect, as in the first one, part of the vapor generated is used to preheat the seawater that flows through the preheaters. Finally, the vapor produced in the last effect is led to the end condenser, where is condensed by transferring the latent heat of evaporation to the cooling seawater which is passing through the condenser bundle tube. That heated seawater is divided into two streams, some is pumped to the first effect of the plant after passing through the preheaters of each effect and the rest is rejected. The vapor condensed in each effect and in the preheaters together with the distillate from the last condenser make up the total distillate of the plant. In order to improve the process from an energetic point of view, the distillate produced in each effect goes to other effects of the plant to recover part of its sensible heat. In some cases, the distillate goes from an effect to the next one, and in other cases, it goes to further effects as follows: in the fourth cell all the distillate is extracted, part of it goes to the seventh cell and the rest to the tenth. Similar extraction is made in the seventh cell, splitting the condensate between the tenth and the thirteenth effects. Another extraction takes place from the tenth cell, part goes to the thirteenth effect and the rest is mixed with the distillate produced in the fourteenth effect. The final extraction is made in the thirteenth effect, after which all the distillate is mixed with that produced in the fourteenth effect. Finally, this accumulated distillate is mixed with the distillate from the condenser, making up the total production. Besides the vapor formed by boiling, a small portion is formed by flashing. When the distillate and the brine pass from one effect to another, some flashing
3
takes place since they enter a cell that is at a lower pressure than the previous one. The vacuum system consists of two hydro-ejectors, which are connected to effects 2, 7, and the end condenser. They are within a closed circuit with a tank and an electric pump that pumps seawater through the ejectors at a pressure of 3 bar. This vacuum system does the initial vacuum of the plant removing the air and the non-condensable gases generated during the desalination process. The design specifications of the PSA MED plant are given in Table 1. The tube bundles of the heater, evaporators, preheater, and condenser are made of 90–10 Cu–Ni tubes. The surface areas of each of the different tube bundles are the following: • • • •
First effect evaporator bundle, 24.26 m2. Effects 2–14 evaporator bundle, 26.28 m2. Preheater bundle, 5 m2. Condenser bundle, 18.3 m2.
2.2. Experimental set-up The MED-PSA plant is experimental and therefore equipped with a comprehensive monitoring system, which provides instantaneous values of the measured data. The monitored data are detailed in Table 2 and are also indicated in Fig. 1. The supply water to the desalination plant is obtained from wells and stored in a pool from where the cooling water is pumped to the bundle tube of the condenser. Afterwards, part of it is used as feed water Table 1 Design specifications of the MED-PSA plant Number of effects Feed seawater flow rate Brine flow rate from the last effect Hot water flow rate Total distillate output Cooling seawater flow rate at 25 ˚C Vapor production in the last effect at 70 ˚C Heat source energy consumption Performance ratio Vacuum system Inlet/Outlet hot water temperature Brine temperature (on the first cell) Feed and cooling seawater temperature at the outlet of the condenser
14 8 m3/h 5 m3/h 12.0 L/s 3 m3/h 20 m3/h 159 kg/h 200 kW >9 Hydro-ejectors (seawater at 3 bar) 74.0/70.0˚C 68˚C 33˚C
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Table 2 Monitored data at PSA MED plant Measurement
Name of variable
Magnitude
Flow rate
Mh Mcw Mf Md Mb
Heating water flow Cooling seawater flow Feed seawater flow Product water flow Brine flow
Temperature
Th,in Th,out Tf Tcw,in Tcw,out
Heating water inlet Heating water outlet 1st effect sprayed seawater temp. Cooling seawater inlet temp. Cooling seawater (rejected) outlet temp.
Pressure
Pv(1), Pv(2), Pv(4), Pv(6), Pv(8), Pv(10), Pv(12), Pv(14) Ph,in Pvc
1st, 2th, 4th, 6th, 8th, 10th, 12th, 14th effect vapor press. Heating water inlet pressure Vapor pressure in the condenser
Salt concentration
Xf
Seawater TDS at the condenser inlet
in the first effect and the rest is rejected back to the pool. Therefore, part of the heat released at the condenser goes into the pool and the cooling water temperature could increase during the experiment, which is not desirable. To avoid that, a dry cooler is switched on since the beginning of the experiment cooling the pool. As the model proposed is at steady-state, time averages of the measured data are carried out in order to perform the modeling. 3. Mathematical model The model is based on the steady-state mass and energy balances, and the heat transfer equations for all the components of the plant, considering the work developed by El-Dessouky and Ettouney (2002). As opposed of the model of El-Dessouky, the first effect has been modeled taking into account that this unit uses hot water as the thermal energy source instead of vapor. The model has been implemented in MATLAB environment accounting for the processes and features of the MED-PSA plant. The following assumptions have been taken into account in order to simplify the analysis: steady-state operation, negligible heat losses to the surroundings, equal temperature difference across the effects, equal temperature difference across the preheaters, salt-free distillate from all the effects, the condensate that leaves each effect is considered as saturated liquid.
To develop the model, the MED system has been divided into three blocks: the first effect, the effects 2 to N, and the end condenser. Within the second block, there are three sub-blocks: the first one, which consider the effects in which the distillate that enters comes from the previous effect and preheater (effects 3, 4, 6, 9, and 12), the effects in which the distillate that enters comes from the previous effect, from the previous preheater and from further effects (effects 7, 10, and 13), and the effects in which the distillate that enters comes from the previous preheater (effects 2, 5, 8, 11, and 14). The first two blocks (the first effect and the effects 2 to N) have two control volumes: the evaporator and the preheater. The evaporator is the part of the effect where the seawater is sprayed over the bundle tube and part of the brine is evaporated. 3.1. The first effect Fig. 2 shows a schematic diagram of the heater, which corresponds to the first effect of the plant. The mass, salt, and energy balances of the evaporator are the following: • Energy balance: Mh hh;in þ Mf hf ¼ mev ð1Þ hev ð1Þ þ Mh hh;out þ mb ð1Þ hb ð1Þ
ð1Þ
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This parameter can be determined by a correlation published by El-Dessouky [19,20], which is as follows: 2
BPEð1Þ ¼ A Xð1Þ þ B Xð1Þ þ C Xð1Þ
3
ð7Þ
A ¼ 8; 325 102 þ 1; 883 104 Tb ð1Þ þ 4; 02 106 Tb ð1Þ2 B ¼ 7; 625 104 þ 9; 02 105 Tb ð1Þ 5; 2 107 Tb ð1Þ
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Fig. 2. Flow diagram of the heater (effect 1).
2
C ¼ 1; 522 104 3 106 Tb ð1Þ 3 108 Tb ð1Þ2
where Mh is heating water mass flow rate from the solar field, Mf is the feed seawater mass flow rate, mev(1) the vapor mass flow rate that leaves the evaporator, mb(1) the brine mass flow rate that leaves the first effect, and h is the enthalpy corresponding to each stream.
where X(1) is the salt weight percentage that leaves the first effect. The above equation is valid over the following ranges: 1 X(1) 16%, 10 Tb(1) 180 ˚C. The heat transfer equation for the evaporator of the first effect can be written as:
• Mass and salt balances: ðTh;in Tb ð1ÞÞ ðTh;out Tf Þ T Tb ð1Þ ln Th;in h;out Tf
ð8Þ
Mf ¼ mb ð1Þ þ mev ð1Þ
ð2Þ
Qh ¼ Ah Uh
Xf Mf ¼ Xð1Þ mb ð1Þ
ð3Þ
where Uh is the overall heat transfer coefficient for the heater, Ah is the first effect evaporator bundle tube, Th,in the heating water inlet temperature and Th,out the heating water outlet temperature. Qh is the thermal power that the hot water transfers to the seawater that is sprayed over the first effect, and it is defined as follows:
where Xf is the feed seawater salinity and X(1) the salinity of the brine that leaves the first effect. The mass and energy balances of the preheater are the following: • Energy balance:
Qh ¼ Mh Cph ðTh;in Th;out Þ
mev ð1Þ hev ð1Þ þ Mf hp ð1Þ ¼ mv ð1Þ hv ð1Þ þ mdph ð1Þ hdph ð1Þ þ Mf hf
ð4Þ
• Mass balance: mev ð1Þ ¼ mv ð1Þ þ mdph ð1Þ
ð5Þ
where mv(1) is the vapor mass flow rate that leaves the first preheater and mdph(1) is the distillate mass flow rate that is generated in the preheater. It is considered that the condensate generated in the preheater and the rest of vapor leave the preheater at a temperature of Tv(1) (it corresponds to the saturation pressure Pv(1)). This temperature is lower that of the boiling brine temperature (Tb(1)) by the boiling point elevation (BPE). Therefore: Tb ð1Þ ¼ Tv ð1Þ þ ðBPEÞ1
ð6Þ
ð9Þ
where Cph is the hot water-specific heat. In the case of the preheater, the heat transfer equation is given by: Qp ð1Þ ¼ Ap Up
ðTv ð1Þ Tf Þ ðTv ð1Þ Tp ð1ÞÞ v ð1ÞTf ln TvTð1ÞT ð1Þ p
ð10Þ
where Up(1) is the overall heat transfer coefficient of the first preheater, Ap is the preheater bundle (it is considered the same for all the preheaters), Tp(1) is the seawater temperature in the inlet of the first preheater and Tf is the seawater temperature in the outlet of the last preheater. Qp(1) is the thermal power that the vapor coming from the evaporator transfer to the seawater flowing through the tubes of the preheater and it is determined by this equation:
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Qp ð1Þ ¼ Mf Cpf Tf Mf Cp ð1Þ Tp ð1Þ
ð11Þ
where Cp(1) and Cpf are the seawater-specific heat at the inlet of the first preheater and at the outlet of the last one, respectively.
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3.2. Effects 2 to N
flow rate that leaves the preheater i 1 and enters the evaporator i, mbr(i) is the brine mass flow rate that is sprayed over the evaporator I bundle tube, mb(i) is the brine mass flow rate that leaves the evaporator i and md(i) is the distillate mass flow rate that leaves the evaporator i. • Mass and salt balances:
As mentioned earlier, within this block there are three sub-blocks. Each sub-block is composed by three components: the flash box, the evaporator, and the preheater. The balances corresponding to the flash box and the preheater are the same for the three blocks, but those corresponding to the evaporator are slightly different. Therefore, in each sub-block, only the evaporator will be analyzed and then the flash box and the preheater will be analyzed for the three sub-blocks.
3.2.1. Effects in which the distillate that enters comes from the previous preheater The effects corresponding to this sub-block are the following: 2, 5, 8, 11 y 14. Fig. 3 shows a schematic diagram of a typical type of this effect. The mass, salt, and energy balances are the following: • Energy balance:
mdph ði 1Þ hdph ði 1Þ þ mv ði 1Þ hv ði 1Þ
mbr ðiÞ ¼ mev ðiÞ þ mb ðiÞ
ð13Þ
mbr ðiÞ Xbr ðiÞ ¼ mb ðiÞ XðiÞ
ð14Þ
where Xbr(i) is the salinity of the brine that is sprayed over the evaporator i bundle tube and X(i) is the brine salinity that leaves the evaporator i. The distillate coming from the preheater i 1, mdph(i 1), joins the condensate generated inside the tubes of the evaporator i, mc(i), which is produced from the condensation of the vapor coming from the effect i 1, mv(i 1). Both distillates form the total distillate in the effect i, md(i), which leaves the effect as saturated liquid at a temperature of Tv(i) (saturation temperature at the pressure Pv(i)): md ðiÞ ¼ mdph ði 1Þ þ mc ðiÞ
ð15Þ
mc ðiÞ ¼ mv ði 1Þ
ð16Þ
þ mbr ðiÞ hbr ðiÞ ¼ mev ðiÞ hv ðiÞ þ mb ðiÞ hb ðiÞ þ md ðiÞ hd ðiÞ
ð12Þ
where mdph(i 1) is the distillate mass flow rate coming from the preheater i 1, mv(i1) is the vapor mass
3.2.2. Effects in which the distillate that enters comes from the previous effect and preheater The effects corresponding to this sub-block are the following: 3, 4, 6, 9, and 12. Fig. 4 shows a schematic diagram of a typical type of this effect. The mass, salt, and energy balances are the following: • Energy balance: mdph ði 1Þ hdph ði 1Þ þ md ði 1Þ hd ði 1Þ þ mv ði 1Þ hv ði 1Þ þ mbr ðiÞ hbr ðiÞ ¼ mev ðiÞ hv ðiÞ þ mb ðiÞ hb ðiÞ þ md ðiÞ hd ðiÞ
ð17Þ
• Mass and salt balances:
Fig. 3. Flow diagram of the effects 2, 5, 8, 11, and 14.
mbr ðiÞ ¼ mev ðiÞ þ mb ðiÞ
ð18Þ
mbr ðiÞ Xbr ðiÞ ¼ mb ðiÞ XðiÞ
ð19Þ
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mdph ði 1Þ hdph ði 1Þ þ md ði 1Þ hd ði 1Þ þ mdiny ðiÞ hdiny ðiÞ þ mv ði 1Þ hv ði 1Þ þ mbr ðiÞ hbr ðiÞ ¼ mev ðiÞ hv ðiÞ þ mb ðiÞ hb ðiÞ þ md ðiÞ hd ðiÞ
ð22Þ
• Mass and salt balances:
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Fig. 4. Flow diagram of the effects 3, 4, 6, 9, and 12.
In this case, the distillate coming from the previous effect, md(i 1), together with the distillate from the preheater i 1, mdph(i 1), joins the condensate generated inside the tubes of the evaporator i, mc(i). Both distillate form the total distillate in this effect, md(i), that, as before, leaves the effect as saturated liquid at a temperature of Tv(i). md ðiÞ ¼ md ði 1Þ þ mdph ði 1Þ þ mc ðiÞ
ð20Þ
mc ðiÞ ¼ mv ði 1Þ
ð21Þ
3.2.3. Effects in which the distillate that enters comes from the previous effect, from the previous preheater and from further effects
mbr ðiÞ ¼ mev ðiÞ þ mb ðiÞ
ð23Þ
mbr ðiÞ Xbr ðiÞ ¼ mb ðiÞ XðiÞ
ð24Þ
In this case, the distillate coming from the previous effect, md(i 1), together with the distillate from the preheater i 1 and with part of the distillate from further effects, mdiny(i 1), joins the condensate generated inside the tubes of the evaporator i, mc(i). The three distillates form the total distillate in this effect, md(i), that, as in the previous cases, leaves the effect as saturated liquid at a temperature of Tv(i). md ðiÞ ¼ md ði 1Þ þ mdph ði 1Þ þ mdiny ðiÞ þ mc ðiÞ
ð25Þ
mc ðiÞ ¼ mv ði 1Þ
ð26Þ
The distillate mass flow rate that enters the effects 7, 10, and 13, mdiny(i), is determined by taking into account that the sum of distillate volumetric flow rate from the previous effect and this one of the distillate that enters the effects 7, 10, and 13 is equal to 0.24 m3/h. This volumetric flow rate ensures that the
The effects corresponding to this sub-block are the following: 7, 10, and 13. Fig. 5 shows a schematic diagram of a typical type of this effect. The mass, salt, and energy balances are the following: • Energy balance:
Fig. 5. Flow diagram of the effects 7, 10, and 13.
Fig. 6. Flow diagram of the extractions and injections of distillate in the effects.
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rest of tubes are enough to get a suitable velocity of the vapor for the condensation process. Fig. 6 shows a flow diagram of the inlet and outlet distillate streams between the fourth and fourteenth effects. Energy and mass balances are shown below. They will be useful to determine the distillate temperature at the outlet of the mixers and the total distillate that leaves the effects 2 to N. The mass and energy balances through the mixers are shown below: Mixer 1:
The (BPE)i is determined by Eq. (7). The heat transfer equation for a typical evaporator i can be written as follows: Qev ðiÞ ¼ Uev ðiÞAev ðiÞ ðTv ði 1Þ Tb ðiÞÞ
ð39Þ
mdmix ð1Þ ¼ mdrest ð7Þ þ md ð7Þ
ð27Þ
where Uev(i) is the overall heat transfer coefficient of a typical evaporator i, and Aev(i) is the evaporator i bundle tube that is the same for all the evaporators from 2 to N. Qev(i) is the thermal power that is transferred from the vapor coming from the previous effect (i 1) to the seawater sprayed over the bundle tube of the effect i. It is given by:
mdrest ð7Þ ¼ md ð4Þ mdiny ð7Þ
ð28Þ
Qev ðiÞ ¼ mv ði 1Þ kv ði 1Þ
mdmix ð1Þ hdmix ð1Þ ¼ mdrest ð7Þ hdrest ð7Þ þ md ð7Þ hd ð7Þ ð29Þ Mixer 2: mdmix ð2Þ ¼ mdrest ð10Þ þ md ð10Þ
ð30Þ
mdrest ð10Þ ¼ mdmix ð1Þ mdiny ð10Þ
ð31Þ
ð40Þ
where kv ði 1Þ is the latent heat of formed vapor at a temperature of Tv(i 1). In the cases of the control volumes flash and preheater, the mass and energy balances are the same for the three sub-blocks, and they are shown below. Flash:
The brine coming from the previous effect, mb(i 1), enters the effect i which is at a lower presmdmix ð2Þ hdmix ð2Þ ¼ mdrest ð10Þ hdrest ð10Þ þ md ð10Þ hd ð10Þ sure and a portion of vapor is formed by flashing, ð32Þ decreasing its temperature from Tb(i 1) to Tb0 ðiÞ. This temperature is higher than the boiling temperature Mixer 3: within the effect i, Tb(i) by the non-equilibrium allowance, which is a measure of the flashing process [5]: ð33Þ mdmix ð3Þ ¼ mdrest ð13Þ þ md ð13Þ Tb0 ðiÞ ¼ Tb ðiÞ þ ðNEAÞi ð41Þ ð34Þ mdrest ð13Þ ¼ mdmix ð2Þ mdiny ð13Þ This parameter can be determined by the following correlation [20]: mdmix ð3Þ hdmix ð3Þ ¼ mdrest ð13Þ hdrest ð13Þ þ md ð13Þ hd ð13Þ ð35Þ Mixer 4: mdmix ð4Þ ¼ mdmix ð3Þ þ md ð14Þ
ð36Þ
mdmix ð4Þ hdmix ð4Þ ¼ mdmix ð3Þ hdmix ð3Þ þ md ð14Þ hd ð14Þ ð37Þ It is considered that the distillate leaves each effect as saturated liquid at the same temperature of the vapor inside the effect, Tv(i). As in the first effect, the temperature of the vapor generated in the effect i, Tv(i), is lower that of the boiling brine temperature in such effect (Tb(i)) by the boiling point elevation (BPE)i: Tb ðiÞ ¼ Tv ðiÞ þ ðBPEÞi
ð38Þ
ðNEAÞi ¼ 33ðDTb ðiÞÞ0;55 =Tv ðiÞ
ð42Þ
where DTb(i) is: DTb ðiÞ ¼ Tb ði 1Þ Tb ðiÞ
ð43Þ
From the flashing evaporation, an amount of vapor is obtained, m0b ðiÞ, that joins in the way to the preheater the vapor generated by boiling, mev(i). The rest nonevaporated brine, mbr(i), is sprayed over the evaporator bundle tube. The energy, mass, and salts balances are shown below. • Energy balance: mb ði 1Þ hb ði 1Þ ¼ mbr ðiÞ hbr ðiÞ þ m0b ðiÞ h0b ðiÞ
ð44Þ
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Qp ðN 1Þ ¼ Mf ðhp ðN 1Þ hcw;out Þ
• Mass and salts balances: mb ði 1Þ ¼ mbr ðiÞ þ m0b ðiÞ
ð45Þ
mb ði 1Þ Xði 1Þ ¼ mbr ðiÞ Xbr ðiÞ
ð46Þ
Preheater: The energy and mass balances are shown below. • Energy balance:
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ð53Þ
The temperature difference across the effects (DTv) and the preheaters (DTp) is calculated by the following expressions: Evaporators: DTv ¼
Tv ð1Þ Tv ðNÞ N1
ð54Þ
Then, the vapor temperature in each evaporator, Tv(i), is determined as follows:
mev ðiÞ hev ðiÞ þ m0b ðiÞ h0b ðiÞ þ Mf hp ði þ 1Þ ¼ mv ðiÞ hv ðiÞ þ mdph ðiÞ hdph ðiÞ þ Mf hp ðiÞ
9
ð47Þ
In the case of the last preheater (i = N 1), the seawater inlet temperature is the seawater outlet temperature from the condenser, Tcw,out, so the energy balance can be written as follows:
Tv ðiÞ ¼ Tv ði 1Þ DTv
ð55Þ
Preheaters: DTp ¼
Tf Tcw;out N1
ð56Þ
Therefore, the seawater temperature in each preheater is determined as follows:
mev ðN 1Þ hev ðN 1Þ þ m0b ðN 1Þ h0b ðN 1Þ þ Mf hcw;out ¼ mv ðN 1Þ hv ðN 1Þ þ mdph ðN 1Þ hdph ðN 1Þ þ Mf hp ðN 1Þ
Tp ðiÞ ¼ Tp ði þ 1Þ þ DTp
ð57Þ
• Mass balance: mev ðiÞ þ m0b ðiÞ ¼ mv ðiÞ þ mdph ðiÞ
ð49Þ
The heat transfer equation for a typical preheater i is written as follows: Qp ðiÞ ¼ Ap ðiÞ Up ðiÞ
ðTv ðiÞ Tp ðiÞÞ ðTv ðiÞ Tp ði þ 1ÞÞ Tv ðiÞTp ðiÞ ln Tv ðiÞT p ðiþ1Þ
ð50Þ
where Up(i) is the overall heat transfer coefficient for a typical preheater, Ap(i) is the preheater bundle of a preheater (it is the same for all the preheaters), and Qp(i) is the thermal power that is transferred from the vapor coming from the evaporator i to the seawater flowing through the preheater bundle tube. It is determined as follows: Qp ðiÞ ¼ Mf ðhp ðiÞ hp ði þ 1ÞÞ
ð51Þ
In the case of the last preheater, the heat transfer equation is given by:
3.3. The end condenser The end condenser (see the flow diagram in Fig. 7) is located next to the last effect of the plant (i = N), so the vapor generated in this effect flows to the condenser through the demister and releases its latent heat to the seawater flowing through the bundle tube. Therefore, the vapor condenses and the seawater increases its temperature. The mass, salt, and energy balances are the following: • Energy balance: m0b ðNÞ h0b ðNÞ þ mev ðNÞ hev ðNÞ þ ðMf þ Mcw Þ hcw;in ¼ ðMf þ Mcw Þ hcw;out þ mdc hvc
ð58Þ
where Mcw is the cooling seawater mass flow rate. Since negligible heat losses to the surroundings have been considered, the vapor temperature inside the effect N, Tv(N), is set to be the same as the vapor temperature in the condenser, Tvc (it corresponds to the saturation pressure, Pvc).
Qp ðN 1Þ ¼ Ap ðN 1Þ Up ðN 1Þ
ðTv ðN 1Þ Tp ðN 1ÞÞ ðTv ðN 1Þ Tcw;out Þ T ðN1ÞT ðN1Þ ln Tv v ðN1ÞTpcw;out
where Qp(N 1) is written as follows:
ð52Þ
• Mass balance: mdc ¼ mev ðNÞ þ m0b ðNÞ
ð59Þ
P. Palenzuela et al. / Desalination and Water Treatment
10
Finally, the product water of the plant, Md, and its temperature, Td, are determined by a mass and energy balance in the Mixer 5: • Mass balance: Md ¼ mdc þ mdmix ð4Þ
ð60Þ
• Energy balance: mdmix ð4Þ hdmix ð4Þ þ mdc hdc ¼ Md hd
ð61Þ
Downloaded by [Guillermo Zaragoza] at 09:43 18 July 2012
On the other hand, the heat transfer equation of the end condenser is given by: Qc ¼ Ac Uc
ðTv ðNÞ Tcw;out Þ ðTv ðNÞ Tcw;in Þ cw;out ln TTvvðNÞT ðNÞTcw;in
ð62Þ
where Uc is the overall heat transfer coefficient of the condenser, Ac is the condenser bundle, and Qc is the thermal power transferred from the vapor coming from the last effect to the feed and cooling seawater flowing through the condenser bundle tube. It is written as follows: Qc ¼ ðMf þ Mcw Þ ðhcw;out hcw;in Þ
ð63Þ
The model described earlier is useful to determine the performance ratio (PR) of the plant, which is defined as kg of distillate produced for every 2,326 kJ of thermal energy supplied to the system. Therefore, the equation to assess it is as follows: PR ¼
Md 2326 kJ 1 kg Qh
4. Parameterization In the equations described earlier, the following are the series of variables: the overall heat transfer coefficient for the first effect (Uh), the overall heat transfer coefficient for the preheaters (Up(i)), the vapor temperature inside the first effect (Tv(1)), and the cooling seawater outlet temperature (Tcw,out), whose value is needed to determine in order to run the model. For this purpose, a series of parametric equations that represent these variables have been obtained from a three-level factorial experimental design (3k), The variables or factors (k) with the actual levels of operation (low (1), medium (0), and high (+1)) are shown in Table 3. Therefore, an experimental campaign have been performed with a total of 81 (34) experiments. The experiments were carried out in 10 months, from March to January, and they were designed taking into account that the last effect vapor temperature has to be higher that the minimum seawater temperature in the Mediterranean Sea (that is 15˚C) plus a temperature difference of 10˚C in the condenser in order to satisfy the cooling flow rates required. The overall heat transfer coefficient for the first effect is obtained from Eq. (8). However, for the pre p is calcuheaters an overall heat transfer coefficient, U lated by considering the total heat transfer, Qp in all N 1 preheaters: p ¼ U
Qp Ap DTp ðN 1Þ
The total heat transfer rate in the N 1 preheaters is assessed as follows: Qp ¼ Mf ðhf hcw;out Þ
ð64Þ
ð66Þ
A second-order response surface model (Response Surface Methodology, RSM) has been used in order to obtain the parametric equations. Each response has been linked to the factors by a second-order polynomial model with interactions as shown in the following equation [21]: Y ¼ bo þ
k X i¼1
Fig. 7. Flow diagram of the end condenser.
ð65Þ
bi Xi þ
k X i¼1
bii Xi2 þ
k1 X X
bij Xi Xj þ e
ð67Þ
i¼1 j¼2j[i
where Y is the response, bo, b1, … , bk, bij are the regression coefficients, Xi, Xj (j = i + 1, … , k) represent the independent variables or factors (last effect vapor temperature, Tv(14), heating water inlet temperature, Th,in, feed seawater flow, Mf, and heating water flow, Mh) and e is the statistical error.
P. Palenzuela et al. / Desalination and Water Treatment
11
Table 3 Variable factors and their actual values of operation
Last effect vapor temperature (˚C) Heating water inlet temperature (˚C) Feed seawater flow (m3/h) Heating water flow (L/h)
Symbol
1
0
+1
Tv(14) Th,in Mf Mh
25 65 6 7
30 70 7 9.5
35 75 8 12
The RSM model coefficients for each response are computed by multiple linear regression (MLR) method ([21,22]):
Downloaded by [Guillermo Zaragoza] at 09:43 18 July 2012
b ¼ ðXT XÞ1 XT Y
ð68Þ
where b is the vector formed by the regression coefficients, X is the matrix (N u) of the independent variables, u is the number of regression coefficients in the RS-model (Eq. (67)) and Y is a vector (N 1) formed by the responses of the N experiments. According to this method, the b coefficients are determined by the method of least squares. In other words, the b values are chosen in order to minimize the sum of squared residuals. For each response variable, the parametric equations with the corresponding regression coefficients have been determined. 5. Results and discussion The independent and dependent variables are fitted to the second-order model equation (Eq. (67)) and for each response variable is examined the goodness of fit. It and the significance of each regression coefficient were obtained using the Modde 5.0 software with a confidence level of 95%. Tables 4 and 5 present the regression relationships for each response monitored and the p values. They are used as a tool to check the significance of each of the coefficients, which in turn may indicate the pattern of the interaction between the variables. The smaller the value of p, the more significant is the corresponding coefficient [23]. The following regression equations represent the best description of the variables Tv(1), Tcw,out, Uh and Up after the elimination of non-significant parameters (p > 0.05) from the results summarized in Tables 4 and 5: Tv ð1Þ ¼ 85:9763 ð1:06873 Tv ðNÞÞ ð1:20385 Th;in Þ ð0:974267 Mf Þ þ ð1:43293 Mh Þ 2 Þ ð0:055331 M2h Þ þ ð0:0104179 Th;in
þ ð0:017083 Tv ðNÞ Th;in Þ
ð69Þ
Tcw;out ¼ 19:8783 þ ð1:27265 Tv ðNÞÞ þ ð0:432938 Th;in Þ ð0:193712 Mh Þ þ 0:00391375 Tv2 ðNÞÞ ð0:00199305 2 Þ þ ð0:00936997 M2h Þ ð0:00632534 Th;in
Tv ðNÞ Th;in Þ
ð70Þ
Uh ¼ 25:1217 ð0:0992825 Tv ðNÞÞ ð0:678212 Th;in Þ þ ð0:30056 Mh Þ ð0:00323972 2 Þ ð0:0112135 Tv2 ðNÞÞ þ ð0:00392379 Th;in
M2h Þ þ ð0:00442078 Tv ðNÞ Th;in Þ
Up ¼ 0:000540399 þ ð0:836569 Mf Þ
ð71Þ
ð72Þ
The goodness of fit of the model is evaluated by the coefficient of determination (R2). It is defined as the proportion of variation in the response attributed to the model. It is suggested that R2 should be closed to 1 for a good fit model [24]. However, a large value of R2 does not always imply that the regression model is a good one. R2 always increases with the addition of a new variable to the model, regardless of weather additional variable is statistically significant or not [25]. Thus, it is preferred to use the adjusted-R2 to evaluate the model adequacy since it is adjusted for the number of terms in the model. The adjusted-R2 should be over 90% indicating a high degree of correlation between the observed and predicted values [25]. Besides these coefficients, another one to evaluate the goodness of fit of the model is the coefficient Q2, which is defined as the fraction of variation of the response that can be predicted by the model. Values of Q2 close to 1 indicate a very good model [26]. Table 6 summarizes the statistics used to test the adequacy of the model. The results indicate that all the fit
P. Palenzuela et al. / Desalination and Water Treatment
12
Table 4 Test of significance for the response variable Tv(1) and Tcw,out regression coefficients
Downloaded by [Guillermo Zaragoza] at 09:43 18 July 2012
Model term
Tv(N) Thin Mf Mh Tv(N) · Tv(N) Thin · Thin Mf · Mf Mh · Mh Tv(N) · Thin Tv(N) · Mf Tv(N) · Mh Thin · Mf Thin · Mh Mf · Mh
Tcw,out
Tv (1) Coefficient estimate
p-Value
Coefficient estimate
p-Value
0.460 1.295 7.725 0.798 0.005 0.011 0.679 0.052 0.015 0.056 0.009 0.060 0.013 0.013
